Lucy Letby was convicted last year of murdering seven babies and attempting to kill six others at the Countess of Chester hospital in north-west England, where she worked as a nurse. She is serving 14 whole-life sentences, meaning she will never be released from prison. In April she sought permission to appeal against those convictions at the court of appeal in London, and in May her challenge was refused. On Tuesday the court published its judgment explaining its decision. Here we explain what steps she has taken – and what could happen next. What stages has Letby’s case been through at the court of appeal? Any defendant wanting to make an appeal against a crown court conviction or sentence must seek permission to do so from the court of appeal criminal division, the second most senior court in England and Wales. This court, based at the Royal Courts of Justice in London, is able to reduce a prison sentence or quash a crown court conviction. It can order a retrial in cases where a judge has made an error. After her trial last year, Letby’s lawyers sought permission to appeal against her convictions. She put forward four grounds of appeal, each of which related to applications she had made during the trial that had been denied by the judge, Mr Justice Goss. On 24 May, judges denied permission. What did she argue at the appeal court? Letby’s legal team argued that her convictions were unsafe because Goss had been mistaken in law when he refused four separate applications made by the defence. First, they said the evidence of the prosecution’s key expert, Dr Dewi Evans, should have been struck out for being “dogmatic and biased”. Letby’s barrister, Benjamin Myers KC, also argued that the medical evidence behind her convictions for fatally injecting air into babies’ bloodstreams, producing what is known as an air embolus, was “very weak” and could not be relied upon. Dr Shoo Lee, an academic who researched the phenomenon of air embolus in babies in 1989, gave evidence to the appeal court on behalf of the defence, telling judges that the “only sign” of it was pink blood vessels “superimposed” on a pink or blue body. When questioned by the prosecution, Lee accepted he had not assessed any of the victims’ medical records in Letby’s case or seen any of the witness testimony provided by other medical staff, who noted strange rashes on a number of the dead babies. Letby’s third argument was that the judge was wrong to direct the jury that they did not have to be sure of the precise act, or acts, that led to a baby’s collapse or death. Finally, Myers said the judge failed to properly investigate a complaint made about a juror in the final days of the trial, when they were allegedly overheard saying they had “made up their minds from the start” about Letby’s guilt. It was later suggested that the complaint may have been made maliciously as the complaint was traced back to a cafe owner who had been in dispute with the juror’s partner over the sale of an iPhone. Why was her application thrown out? In a 59-page ruling published on Tuesday, the appeal court judges said Goss’s handling of the trial had been “thoughtful, fair, comprehensive and correct”. They ruled that none of the four legal challenges advanced by Letby were “arguable” and said they did not consider that the criteria for the admission of fresh evidence – in the form of two research papers by Lee – had been met. Is this the end of the road for Letby’s appeal? What could she do next? After her initial application for leave to appeal was rejected, Letby renewed her application, essentially putting in the appeal application for a second time, to be reconsidered by a panel of judges at a hearing. When this was denied, in effect she had exhausted the appeals process. One avenue still available to Letby may be the Criminal Cases Review Commission (CCRC). What does the Criminal Cases Review Commission do? The CCRC is an official public body tasked with investigating potential miscarriages of justice in England, Wales and Northern Ireland. It can refer a case back to the court of appeal if it identifies new or overlooked evidence. It has the legal power to obtain evidence from public bodies, such as the police and the Crown Prosecution Service, and private organisations. It can also interview witnesses and arrange for new expert evidence such as psychological reports and DNA testing. For the CCRC to be able to refer a case back to the court of appeal, there would need to be new information that may have changed the outcome of the case if the jury had known about it.